The Bloch-Siegert shift is a well-known perturbation to the rotating-wave approximation which becomes prominent when the nutation frequency of the driving rf field becomes comparable in magnitude with the Larmor frequency of the driven spins.
Consider the following scenario: we are equipped with a set of B1 coils at low-field (the longitudinal B0 field is~1.1 mT). Our untuned B1 coils are capable of broadband irradiation of spins with a transverse B1 field whose nutation frequency, ω1/(2π), can reach around ~2 kHz*.
A 2 kHz nutation frequency presents little food-for-thought for 1H spins (ω0/(2π) @ 1.1 mT ≈ 47 kHz) but really makes you reconsider your life choices if, like us, you have been working with 103Rh (ω0/(2π) @ 1.1 mT ≈ 1.5 kHz).
In this scenario it is necessary to calculate higher-order terms of the Bloch-Siegert shift Hamiltonian, which I have done to 12th (or 11th, depending on your convention) order.
*there is a subtle point here: with untuned coils operating in the high-inductance limit, the B1 field strength is proportional to 1/(ω0), whereas the nutation frequency is of course proportional to ω0. These factors cancel out as shown in pages 90-91 [Figs. 39-40 of my thesis], leading to a nutation frequency that is independent of frequency and hence γ:
In 2016 a Master’s student at Ulm’s Institute of Theoretical Physics, by the name of Benedikt Tratzmiller, described a simple yet powerful control sequence for optical DNP in diamond NV centres. The pulse sequence had the attractive name of PulsePol. The sequence made further appearances in an excellent paper and Tratzmiller’s PhD thesis.
Some time ago I listened to an inspiring online talk by Nino Wili (who quite convincingly spoke about cross-pollination between different fields of magnetic resonance) and we got to talking. After a few discussions/simulations eventually we realized that the PulsePol sequence could be applied to singlet NMR, and that application forms the basis of our paper, which chose to explain PulsePol using the language of symmetry-based sequence design borrowed from solid-state NMR.
PulsePol has some advantages over the M2Ssequence, which was essentially a “default” option in NMR groups for generating nuclear singlet order (alongside variations of the arguably more elegant SLIC method invented by DeVience, Walsworth, and Rosen). These advantages include superior robustness (PulsePol is generally less sensitive to rf errors than M2S), simplicity, and even a small time advantage (PulsePol is ~1.21x faster than M2S). One should expect PulsePol to replace M2S in the future.
Here I provide some ready-to-use pulse programs, written for Bruker TopSpin, that the NMR community may use to actually implement the PulsePol sequence.
The pulse programs have a number of features:
The so-called “riffling” 180 phase shift modification on the central 180° pulses, which improves robustness against pulse strength (Rabi frequency) and resonance offset (detuning) errors.
A T00 gradient filter for selective filtration of singlet order, and a z-filter to select longitudinal magnetization.
A singlet order destruction (SOD) element before the relaxation delay to purge residual singlet order, which may interfere with experiments.
Wimperis’ BB1 composite pulse (in the symmetrized implementation) replacing the pulses in the filters, improving robustness against pulse strength errors.
I also attach a quick reference table which provides the 2 experimental parameters that are actually relevant for optimal excitation of singlet order, because it has always annoyed me that the first one was not explicitly stated in our somewhat cryptic original paper:
Note that I have given the total duration of the PulsePol in terms of the SLIC duration 1/(√2 Δ) – which is the fastest currently known way to fully excite singlet order from longitudinal magnetization. The minimum of the total duration is around n = 3, N = 4, where the total evolution is ~1.38x longer than the SLIC sequence. The (fixed) total duration of the M2S sequence (~1.67x longer than SLIC) is given for comparison.
The “default” sequence for most people working with nearly equivalent spin-1/2 pairs would be R431 – or possibly R873, which is only ~3.5% slower than R431 but provides improved robustness against resonance offset/detuning errors. However, in the common case where a spin system is at intermediate inequivalence (the chemical shift has a comparable magnitude with respect to the J coupling), one can benefit from the sequences in the series R411, R612, R813, R1014… which perform over an increasingly wider range of inequivalence angles at the expense of being increasingly slower, as I discussed in the appendix to this paper.
Most nuclei in the periodic table that possess spin are quadrupolar (i.e. spin ℓ > 1/2). It is well-known that, in the solution-state, a J-coupling between the I (ℓ = 1/2) and S (ℓ >1/2) spins “vanishes” (or more precisely induces a scalar relaxation effect) in the spectrum of the I spin(s) when the following condition is satisfied:
That is to say, J-couplings to quadrupolar spins are not directly observable when the rate of relaxation (1/T1) of the S spins significantly exceeds the J-coupling. Sometimes this is referred to as “self-decoupling”, a term coined by Spiess, Haeberlen, and Zimmermann.
But there are exceptions. The relaxation of the S spins is usually overwhelmingly dominated by the quadrupolar mechanism, whose contribution (in the extreme narrowing limit) is given by:
Here, the norm of the quadrupolar coupling tensor is related to the quadrupolar coupling constant (QCC):
And the quadrupolar coupling constant itself is:
Here, it is worth explaining why I have written the equation in an unconventional way. The nuclear quadrupole moment Q (an intrinsic nuclear property which is non-zero for ℓ > 1/2) is proportional to the eccentricity of the spheroid charge distribution unique to each nuclear species, and has units of area. On the other hand, Vzz (in units voltage/area) represents the principal component of the electric field gradient (EFG) tensor, and EFGs may technically be present even at spin-0 or spin-1/2 nuclei. It is clear that Q and Vzz couple to produce a voltage at the nucleus. Finally, half the Josephson constant (½KJ, in units of frequency/voltage) is the fundamental quantum of voltage-driven oscillation. This basic physical picture is rarely appreciated; a run-of-the-mill QCC of ~150 kHz corresponds to a ~100 picovolt potential difference at the nucleus.
Now it is simple to figure out the exceptions when quadrupolar relaxation may be slow enough to observe J-couplings:
Quadrupolar spins with an intrinsically small nuclear quadrupole moment Q. Examples include 2H and 17O.
Molecules where the quadrupolar spin experiences a small electric field gradient V. Examples are molecules with intrinsically high symmetry, such as 14NH4+ , but it is worth noting that this criterion may not be sufficient for spins where Q is massive.
A larger heteronuclear J-coupling, which is increasingly possible when the I and/or S spin(s) has a larger atomic number and gyromagnetic ratio. But again, the J-coupling must be “large” relative to the quadrupolar relaxation rate.
Now, suppose you actually observe, in the I spectrum, a well-resolved coupling to a single quadrupolar spin S. It is well-known that one would observe 2S+1 Lorentzian peaks with equal areas/integrals. It is less well-known that the Lorentzian linewidths of these peaks would not be equal:
This remarkable effect was described by the Nobel laureate John Pople in 1958, as well as Masuo Suzuki & Ryogo Kubo in 1963, but it appears that the name “Pople-Suzuki-Kubo effect” never really caught on.
Prominent examples where the “PSK effect” have been observed include the hexafluorides, e.g. of niobium (93Nb, I=9/2), antimony (121/123Sb with I=5/2 and 7/2 respectively), or bismuth (209Bi, I=9/2). Insanely enough, there are even examples involving the unpleasant nucleus 235U (I=7/2).
However, my favorite example (and I must admit some personal bias) is certainly the fine paper by Stuart J. Elliott et al. from the Levitt group. The paper not only shows a beautiful spectrum:
But impressively, the paper also provides a hidden gem in the supporting information: a general expression for the linewidths that was derived with the help of the commutator relations of spherical tensor operators. The expression may be written:
And I have provided a triangle of the linewidth coefficients k(m) [note that the intensities would be provided by 1/k(m)] here:
Archetype (noun); the original pattern or model of which all things of the same type are representations or copies.
The Merriam-Webster Dictionary
In many spin dynamical problems, I like to say there is an “archetypal” pulse sequence. The archetypal pulse sequence may be defined as the simplest possible way of achieving a unitary transformation from Operator A to Operator B. Typically the relevant operators are spin-state populations, so the behaviour expected under these sequences would correspond to the classic scenario of Rabi oscillations (i.e. motion from Population A to Population B, and back again). A recurrent motif in pulse sequence development is that the archetypal pulse sequence consists of nothing more than simple cw irradiation (perhaps preceded by a phase-shifted 90 pulse), whose amplitude is adjusted to satisfy a particular matching condition.
Examples of archetypal pulse sequences include:
1. The classic Hartmann-Hahn sequence for polarization transfer, which exchanges populations between spin(s) I and spin(s) S. The matching condition of the Hartmann-Hahn sequence is that the nutation frequency of the simultaneous driving field on the I and S spins coincides. 2. The NOVEL (nuclear orientation via electron spin-locking) sequence for DNP, which exchanges populations between electron (e) and nuclear (n) spins. The matching condition of NOVEL is that the nutation frequency of the driving field on the electron spins coincides with the Larmor frequency of the nuclear spins. 3. The SLIC (spin-lock induced crossing) sequence from the field of singlet NMR, which exchanges populations between (either of) the outer triplet states and the nuclear singlet state. The SLIC matching condition is that the nutation frequency of the driving field applied to a spin pair coincides with their homonuclear J-coupling.
All these pulse sequences are not that different:
This interpretation is quite useful. The archetypal pulse sequences are a direct map between fields of quantum dynamics that appear totally different at first sight. It immediately allows us to translate pulse sequences that were developed in one context, into another. Irrespective of the minutiae of the actual target effective Hamiltonian, and whatever populations are actually being exchanged, we can see that pulse sequences developed for, say, singlet NMR, may be adapted for DNP purposes, and vice versa.
In addition, archetypal pulse sequences possess interesting properties. For example, the SLIC sequence sets the bound on the fastest possible transfer from longitudinal magnetization to nuclear singlet order, which is just:
As a pulse sequence designer, I (and the previous authors, evidently) think this is how we should be teaching these sequences to the coming generations of magnetic resonance. And I think that people in different fields (DNP, ssNMR, NV centres, solution-state singlet NMR, etc.) should talk to each other.
Welcome to my blog. Its purpose is to share information I have stumbled upon during my research that may be useful to members of the NMR community, relating to pulse sequence development, experimental implementation, and cool theoretical aspects of quantum dynamics.
